White Sox Open Checkbook After Once Being Tight With Bullpen Budget

GLENDALE, Ariz. — The strategy sounds wise, because each point behind it is sound. Relief pitchers are notoriously unpredictable. With shrewd scouting and coaching, teams can find effective, low-cost bullpen arms. Therefore, it is foolish to spend big on relievers.

The Chicago White Sox, perhaps better than any other team, proved this theory for years. They used three closers on their way to the 2005 World Series title, with the last, Bobby Jenks, coming on a waiver claim. After letting Jenks leave as a free agent, they used three closers in four seasons. None made more than $520,000 per year.

“It sorts itself out if you’re watching,” Don Cooper, Chicago’s longtime pitching coach, said while watching a practice last spring. “But you’ve got to have a plan. The plan can’t be just to wing it.”

The White Sox tried to wing it last season. They crashed. The team led the majors in bullpen losses, with 32, and ranked 28th in earned run average, at 4.38. The plan, Cooper said, seemed to change from week to week.

“You’re trying to win games, and when you’re not confident in the seventh, eighth and ninth, it just deflates your team,” Manager Robin Ventura said. “They work so hard at scoring the runs and getting the lead early on. If you blow it late, and if they don’t feel that they can win consistently, it just sucks the life out of them. It’ll drive you crazy.”

His new bullpen, Ventura said, is a lot simpler. It is also a lot more expensive, but the White Sox believe it is worth it. The former Yankees closer David Robertson will pitch the ninth after signing a four-year, $46 million contract. Setting him up on many nights will be the left-hander Zach Duke, who signed a three-year, $15 million deal after a stellar season for Milwaukee.

The pitchers are part of a swift makeover by the White Sox, who went 73-89 last season and also acquired starter Jeff Samardzija, outfielder Melky Cabrera, the first baseman and designated hitter Adam LaRoche and infielder Emilio Bonifacio.

“I’ve been here for 28 years,” Cooper said. “There’s never been a year we’ve made this many bulk changes to change the lay of the land for us.”

Without Duke and Robertson, though, the other additions might not matter much. To help recruit the relievers, General Manager Rick Hahn turned to Cooper, who lives near Duke in Tennessee. Cooper set up lunch at Merchants, an upscale restaurant in Nashville, and brought an iPad to analyze Duke’s highlights.

“I had no idea any of that was going to be going on during lunch,” Duke said. “It caught me off-guard a little bit, but it proved to me the White Sox had done their research and were really committed to putting a winning product back on the field.”

Duke signed before Thanksgiving, and while Cooper shared an encouraging phone call with Robertson, signing him still seemed far-fetched. Most of the industry — Robertson included — had no idea that the Yankees did not want to bring him back.

When the Yankees like a player, that player tends to “stay with the Yankees, yeah,” Robertson said, completing the premise of the question. “I knew going into free agency that if I got to that point, the opportunity to leave the Yankees would be there. But I didn’t know what I was going to do.”

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The left-hander Zach Duke signed a three-year, $15 million deal with the White Sox after a stellar season for Milwaukee.CreditLisa Blumenfeld/Getty Images

The Yankees liked the idea of signing the free-agent reliever Andrew Miller, who would not cost them a first-round draft pick because he had been traded in mid-2014.

If they did that, the Yankees, eager to improve their farm system, could then let Robertson go and receive a high draft choice as compensation.

That is how it played out, and while Robertson never expected the White Sox to pursue him, he was happy they did. His wife, Erin, he said, has always loved visiting Chicago, and he likes pitching at U.S. Cellular Field, despite a 6.43 E.R.A. in nine games there.

Robertson was so affable with New York reporters that he won their Good Guy Award. But life outside the Yankees’ glare, he said, is just fine. He can even wear a beard.

“It’s a very relaxed atmosphere here, and I don’t know if that’s because of the coaching staff and the group of guys, or if it’s because there’s only five or six media around,” he said, smiling. “I’m enjoying it. You get the same work in, it’s just different.”

Duke said he was impressed by Robertson’s demeanor, the way nothing ever seems to bother him, a trait that served Robertson well in his brief time as the successor to Mariano Rivera.

Robertson, though, was used to the relief routine, with closing experience going back to college. Duke was a starter as recently as 2012. He made 159 starts for Pittsburgh from 2005 through 2010. That season, he was 8-15 with a 5.72 E.R.A., and a trade to Arizona did not improve things.

Houston released him in 2012, and Washington did so in 2013. Duke needed to adjust to the bullpen, quickly, to keep a job.

“As my wife told me, ‘The opportunity to start at the big league level is just not there anymore,’ ” Duke said. “I needed to be open to embracing something new. If I wanted to be here, it was going to have to be in a different role.”

Cincinnati gave Duke his first extended chance in 2013, and he thrived at Class AAA and in a 14-game stint in the majors. He signed with the Brewers and had a 2.45 E.R.A., with 74 strikeouts in 582/3 innings, baffling hitters with a sidearm angle, a more conventional angle, and four pitches — two more than many relievers use.

Duke said he had felt good this spring, despite mixed results, but Robertson laughed off the same question last weekend. He had failed to finish the inning in his previous outing, and when Ventura told him he was taking him out, Robertson said, “I wish you would have earlier.”

Off the field, Robertson said, he is hoping to use the new contract to buy a thousand acres in Alabama for a family hunting lodge. He said that his foundation, High Socks for Hope, has recently been helping veterans find jobs and readjust to life at home.

One problem, though: Robertson may not show those high socks on the field, at least not in April. Fashion could be a hazard of his new town.

“It’s comfortable, but you might see some pants down a little bit,” Robertson said. “I hear it’s so cold in Chicago in the beginning of the season.”

A version of this article appears in print on , on Page SP3 of the New York edition with the headline: Once on Bullpen Budget, White Sox Open Wallet. Order Reprints | Today’s Paper | Subscribe